University Question Paper Solution

Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)= e x(t) [Dec 12, 10 marks] Solution:- 2. Distinguish between: i) Deterministic and random signals and ii) Energy and periodic signals. [Dec 12, 6 marks] Solution;- Dept of ECE/SJBIT Page 1

3. For any arbitrary signal x(t) which is an even signal, show that [Dec 12, 4 marks] Solution:- 4. Distinguish between: [Dec 09, 8 marks] i) Continuous time and discrete time signals ii) Even and odd signals iii) Periodic and non-periodic signals iv) Energy and power signals. i) Continuous-Time and Discrete-Time Signals A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since a discrete-time signal is defined at discrete times, a discrete-time signal is often identified as a sequence of numbers, denoted by {x,) or x[n], where n = integer. Illustrations of a continuous-time signal x(t) and of a discrete-time signal x[n] are shown in Fig. 1-2. 1.2 Graphical representation of (a) continuous-time and (b) discrete-time signals ii) Even and Odd Signals A signal x ( t ) or x[n] is referred to as an even signal if Dept of ECE/SJBIT Page 2

x (- t) = x(t) x [-n] = x [n] -------------(1.3) A signal x ( t ) or x[n] is referred to as an odd signal if x(-t) = - x(t) x[- n] = - x[n]--------------(1.4) Examples of even and odd signals are shown in Fig. 1.3. 1.3 Examples of even signals (a and b) and odd signals (c and d). Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd. That is, Where, -------(1.5) -----(1.6) Dept of ECE/SJBIT Page 3

Similarly for x[n], -------(1.7) Where, --------(1.8) Note that the product of two even signals or of two odd signals is an even signal and that the product of an even signal and an odd signal is an odd signal. ii) Periodic and Nonperiodic Signals A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive nonzero value of T for which (1.9) An example of such a signal is given in Fig. 1-4(a). From Eq. (1.9) or Fig. 1-4(a) it follows that ---------------------------(1.10) for all t and any integer m. The fundamental period T, of x(t) is the smallest positive value of T for which Eq. (1.9) holds. Note that this definition does not work for a constant signal x(t) (known as a dc signal). For a constant signal x(t) the fundamental period is undefined since x(t) is periodic for any choice of T (and so there is no smallest positive value). Any continuous-time signal which is not periodic is called a nonperiodic (or aperiodic) signal. Periodic discrete-time signals are defined analogously. A sequence (discrete-time signal) x[n] is periodic with period N if there is a positive integer N for which.(1.11) Dept of ECE/SJBIT Page 4

An example of such a sequence is given in Fig. 1-4(b). From Eq. (1.11) and Fig. 1-4(b) it follows that..(1.12) for all n and any integer m. The fundamental period N o of x[n] is the smallest positive integer N for which Eq.(1.11) holds. Any sequence which is not periodic is called a nonperiodic (or aperiodic sequence. Note that a sequence obtained by uniform sampling of a periodic continuous-time signal may not be periodic. Note also that the sum of two continuous-time periodic signals may not be periodic but that the sum of two periodic sequences is always periodic. iii) Energy and Power Signals Consider v(t) to be the voltage across a resistor R producing a current i(t). The instantaneous power p(t) per ohm is defined as (1.13) Total energy E and average power P on a per-ohm basis are (1.14) For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is defined as (1.15) The normalized average power P of x(t) is defined as (1.16) Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is Dept of ECE/SJBIT Page 5

defined as (1.17) The normalized average power P of x[n] is defined as (1.18) 5. Write the formal definition of a signal and a system. With neat sketches for illustration, briefly describe the five commonly used methods of classifying signals based on different features. [July10, 12 marks] Solution:- Signal definition A signal is a function representing a physical quantity or variable, and typically it contains information about the behaviour or nature of the phenomenon. For instance, in a RC circuit the signal may represent the voltage across the capacitor or the current flowing in the resistor. Mathematically, a signal is represented as a function of an independent variable t. Usually t represents time. Thus, a signal is denoted by x(t). System definition A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal.let x and y be the input and output signals, respectively, of a system. Then the system is viewed as a transformation (or mapping) of x into y. This transformation is represented by the mathematical notation y= Tx -----------------------------------------(1.1) where T is the operator representing some well-defined rule by which x is transformed into y. Dept of ECE/SJBIT Page 6

1.1 System with single or multiple input and output signals Classification of signals Basically seven different classifications are there: Continuous-Time and Discrete-Time Signals Analog and Digital Signals Real and Complex Signals Deterministic and Random Signals Even and Odd Signals Periodic and Nonperiodic Signals Energy and Power Signals 6. Explain the following properties of systems with suitable example: i) Time invariance ii) Stability iii) Linearity. [June/July09, 6 marks] Solution:- Time invariance: A system is time invariant, if its output depends on the input applied, and not on the time of application of the input. Hence, time invariant systems, give delayed outputs for delayed inputs. Dept of ECE/SJBIT Page 7

Stability A system is stable if bounded input results in a bounded output. This condition, denoted by BIBO, can be represented by:.(1.42) Hence, a finite input should produce a finite output, if the system is stable. Some examples of stable and unstable systems are given in figure 1.21 Dept of ECE/SJBIT Page 8

Fig 1.21 Examples for system stability Linearity: The system is a device which accepts a signal, transforms it to another desirable signal, and is available at its output. We give the signal to the system, because the output is s Dept of ECE/SJBIT Page 9

7. A continuous-time signal x ( t ) is shown in Fig. 1-17. Sketch and label each of the following signals. ( i ) x(t - 2); ( ii) x(2t); ( iii ) x ( t / 2 ) ; (iv)l x ( - t ) [Jan 05, 10 marks] Solution: 8. Given the signal x(t) as shown in fig 1.b, sketch the following: [Jan/Feb 05, 4 marks] i) X(2t+3) ii) X(t/2-2) Dept of ECE/SJBIT Page 10

Solution:- 9. Find the even and odd components of x (t) = e jt. [Jan 05, 10 marks] Solution: Dept of ECE/SJBIT Page 11

10. Show that the product of two even signals or of two odd signals is an even signal and that the product of an even and an odd signai is an odd signal. Solution: and x( t) is odd. Note that in the above proof, variable I represents either a continuous or a discrete variable. Dept of ECE/SJBIT Page 12

2: Time-domain representations for LTI systems 1 1. Find the convolution integral of x(t) and h(t), and sketch the convolved signal, x(t) = (t -1){u(t -1) +u(t -3)} and h(t)= [u(t +1) - 2u(t -2)]. [Dec 12, 12marks] Solution: Dept of ECE/SJBIT Page 13

2. Determine the discrete-time convolution sum of the given sequences.x(n) ={1, 2, 3, 4} and h(n)= {1, 5, 1} [Dec 12, 8marks] Solution:- 3. Explain any four properties of continuous and / or discrete time systems. Illustrate with suitable examples. [June 7, 8marks] Solution:- Stability A system is stable if bounded input results in a bounded output. This condition, denoted by BIBO, can be represented by:.(1.42) Hence, a finite input should produce a finite output, if the system is stable. Some examples of stable and unstable systems are given in figure 1.21 Memory The system is memory-less if its instantaneous output depends only on the current input. In memory-less systems, the output does not depend on the previous or the future input. Examples of memory less systems: Invertibility: A system is invertible if, Dept of ECE/SJBIT Page 14

A system is causal, if its output at any instant depends on the current and past values of input. The output of a causal system does not depend on the future values of input. This can be represented as: y[n] F x[m] for m n For a causal system, the output should occur only after the input is applied, hence, x[n] 0 for n 0 implies y[n] 0 for n 0 All physical systems are causal (examples in figure 7.5). Non-causal systems do not exist. This classification of a system may seem redundant. But, it is not so. This is because, sometimes, it may be necessary to design systems for given specifications. When a system design problem is attempted, it becomes necessary to test the causality of the system, which if not satisfied, cannot be realized by any means. Hypothetical examples of non-causal systems are given in figure below. 4. What do you mean by impulse response of an LTI system? How can the above be interpreted? Starting from fundamentals, deduce the equation for the response of an LTI system, if the input sequence x(n) and the impulse response are given. [June 7, 7marks] Solution:- The impulse response of a continuous time system is defined as the output of the system when its input is an unit impulse, δ (t ). Usually the impulse response is denoted by h(t ) Figure 2: The impulse response of a continuous time system Dept of ECE/SJBIT Page 15

Evaluation from Z-transforms: Another method of computing the convolution of two sequences is through use of Z-transforms. This method will be discussed later while doing Z-transforms. This approach converts convolution to multiplication in the transformed domain. Evaluation from Discrete Time Fourier transform (DTFT): It is possible to compute the convolution of two sequences by transforming them to the frequency domain through application of the Discrete Fourier Transform. This approach also converts the convolution operator to multiplication. Since efficient algorithms for DFT computation exist, this method is often used during software implementation of the convolution operator. Dept of ECE/SJBIT Page 16

Evaluation from block diagram representation: While small length, finite duration sequences can be convolved by any of the above three methods, when the sequences to be convolved are of infinite length, the convolution is easier performed by direct use of the convolution sum. 5. The input x ( t ) and the impulse response h ( t ) of a continuous time LTI system are given by i)compute the output y ( t ) by Eq ii) Compute the output y ( t ) Solution:- Dept of ECE/SJBIT Page 17

6. Compute the output y(t for a continuous-time LTI system whose impulse response h ( t ) and the input x(t) are given by Solution:- Dept of ECE/SJBIT Page 18

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7. Consider the RL circuit shown in Fig. 2-33. Find the differential equation relating the output voltage y ( t ) across R and the input voltage x( t ). i) Find the impulse response h ( t ) of the circuit. ii) Find the step response s( t ) of the circuit. Solution:- 8. Is the system described by the differential equation Solution:- No, it is nonlinear 9. Write the input-output equation for the system shown in Solution:- 2y[n] - y[n 1] = 4x[n] + 2x[n 1] Dept of ECE/SJBIT Page 20

10. Consider a continuous-time LTI system described by ( a ) Find and sketch the impulse response h ( t ) of the system. ( b ) Is this system causal? Solution:- Dept of ECE/SJBIT Page 21

Unit 3: Time-domain representations for LTI systems 2 1. Determine the condition of the impulse response of the system if system is, i) Memory less ii) Stable. [Dec12, 6marks] Solution:- 2. Find the total response of the system given by, Solution:- Dept of ECE/SJBIT Page 22

3. The system shown in Fig. is formed by connecting two systems in cascade. The impulse responses of the systems are given by h, ( t ) and h 2 ( 0, respectively, and ( a ) Find the impulse response h ( t ) of the overall system shown in Fig. ( b ) Determine if the overall system is BIBO stable. [Dec 11, 10 marks] Solution:- Dept of ECE/SJBIT Page 23

4. Verify the following [Dec 11, 10 marks] Solution:- Dept of ECE/SJBIT Page 24

5. Show that [Jun 10, 10 marks] Solution:- 6. [Jan 10, 10 marks] Dept of ECE/SJBIT Page 25

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7. Find the impulse response h [ n ] for each of the causal LTI discrete-time systems satisfying the following difference equations and indicate whether each system is a FIR or an IIR system. [July05, 10 marks] Dept of ECE/SJBIT Page 27

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Unit 4: Fourier representation for signals 1 1. One period of the DTFS coefficients of a signal is given by x(k)= (1/2) k, on 0< K< 9. Find the time-domain signal x(n) assuming N = 10. [Dec 12, 6marks] Solution: 2. Prove the following properties of DTFs: i) Convolution ii) Parseval relationship iii) Duality iv) Symmetry. [Dec 12, 6marks] Solution: Convolution: As in the case of the z-transform, this convolution property plays an important role in the study of discrete-time LTI systems. Duality: The duality property of a continuous-time Fourier transform is expressed as There is no discrete-time counterpart of this property. However, there is a duality between the discrete-time Fourier transform and the continuous-time Fourier series. Let Dept of ECE/SJBIT Page 29

Since X(t) is periodic with period To = 2 π and the fundamental frequency ω 0 = 2π/T 0 = 1,Equation indicates that the Fourier series coefficients of X( t) will be x [ - k ]. This duality relationship is denoted by Where F S denotes the Fourier series and C k, are its Fourier coefficients. Parseval's Relations: 3. State and prove tbe periodic time shift and periodic properties of DTFS.[ May/June 10 6 marks] Solution: Periodicity As a consequence of Eq. (6.41), in the discrete-time case we have to consider values of R(radians) only over the range0 < Ω < 2π or π < Ω < π, while in the continuous-time case we have to consider values of 0 (radians/second) over the entire range < ω <. Time Shifting: \ Dept of ECE/SJBIT Page 30

4. Give the significance of time and frequency domain representation of signals.give examples..[ May/June 10, 4 marks] 5. If FS representationof a signal x(t0) is X[k]. Derive the FS of a signal x(t-t 0 ).[ Dec09/Jan 10, 6 marks] Solution: Time Shifting: Dept of ECE/SJBIT Page 31

6. Find the inverse Fourier transform x [ n ] Solution: 7. Determine the discrete Fourier series representation for each of the following sequences: Solution: Dept of ECE/SJBIT Page 32

8. Find the Fourier transform of x [ n ] = - a n u[-n-1] Solution: 9. Verify the convolution theorem that is, Solution: Dept of ECE/SJBIT Page 33

10. Using the convolution theorem, find the inverse Fourier transform x [ n ]. Solution: Dept of ECE/SJBIT Page 34

UNIT 5: Fourier representation for signals 2 1. Define the DTFT of a signal. Establish the relation between DTFT and Z transform of a signal. [Jan/Feb 04, 7marks] Solution: Dept of ECE/SJBIT Page 35

2. State and prove the following properties of Fourier transform. i) Time shifting property ii) Differentiation in time property iii) Frequency shifting property: [July 09, 9marks] Time Shifting: Frequency Shifting: Differentiation in Frequency: 3. Show that the real and odd continuous time non periodic signal has purely imaginary Fourier transform. (4 Marks) [Jan/Feb 05, 4marks] Solution: Dept of ECE/SJBIT Page 36

4. Use the equation describing the DTFT representation to determine the time-domain signals corresponding to the following DTFTs : [July 06, 8marks] i) X(e jω )= Cos(Ω)+j Sin(Ω) ii) X(e jω )={1, for π/4<ω< 3π/4; 0 otherwise and X(e jω )=-4 Ω Solution: i) ii) 5. Explain the reconstruction of CT signals implemented with zero-order device. [Jan/Feb 05, 4marks] Solution: 6. Find the DTFT of the sequence x(n) = α n u(n) and determine magnitude and phase spectrum. Solution; Dept of ECE/SJBIT Page 37

7. Plot the magnitude and phase spectrum of x(t) = e -at u(t) Solution; 8. Find the inverse Fourier transform of the spectra, [July 07, 8marks] Solution: Dept of ECE/SJBIT Page 38

9. Find the DTFT of the sequence x(n) =(1/3)n u(n+2) and determine magnitude and phase spectrum. [July 08, 6marks] Solution: 10. Use the defining equation for the FT to evaluate the frequency-domain representations for the following signals: [July 06, 6marks] i) X(t)= e -2t u(t-3) ii) X(t)=e -4t Sketch the magnitude and phase spectra. Solution: i) ii) Dept of ECE/SJBIT Page 39

11. Use the de.ning equation for the DTFT to evaluate the frequency-domain representations or the following signals. Sketch the magnitude and phase spectra. Solution: : [July 07, 8marks] Dept of ECE/SJBIT Page 40

UNIT 6:Applications of Fourier representations 1. December 2011 Solution : 2. December 2011 Solution : 3. December 2012 Dept of ECE/SJBIT Page 41

Solution : 4. December 2012 Solution : Dept of ECE/SJBIT Page 42

5. December 2012 Solution : 6. Use the equation describing the DTFT representation to determine the time-domain signals corresponding to the following DTFT s. [May/Jun 10, 8marks] Solution : Dept of ECE/SJBIT Page 43

7. Use the equation describing the FT representation to determine the time-domain signals correspondingto the following FT s. [July 07, 8marks] Solution: 8. Determine the appropriate Fourier representation for the following time-domain signals, using the defining equations. [July 6, 10marks] Solution: Dept of ECE/SJBIT Page 44

UNIT 7:Z-Transforms 1 1. December 2011 2. December 20112 Solution: 3. December 2012 Dept of ECE/SJBIT Page 45

Solution: 4. December 2012 Dept of ECE/SJBIT Page 46

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UNIT 8: Z-Transforms II 1. December 2012 Solution : 5. December 2012 Solution: Dept of ECE/SJBIT Page 50

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A causal system has input x[n] and output y[n]. Use the transfer function to determine the impulse response of this system. Dept of ECE/SJBIT Page 52

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